# Lab: Introduction to Logic
## Problem 1: Back and Forth
a.
_p~1~_=“It rained this morning.”
_p~2~_=“It will be sunny this afternoon.”
_p~1~_ $\wedge$ _p~2~_

b.
_p~1~_=“I am forced to stay in-doors this week."
_p~2~_=“I will play Genshin Impact."
_p~3~_=“I will have fun."
_p~4~_="I will become poor."
_p~1~_$\rightarrow$_p~2~_ $\wedge$ _p~3~_ $\wedge$ _p~4~_

c.
_p~1~_=“I eat Taco Bell."
_p~2~_=“I eat McDonalds."
_p~3~_=“I will become swoll."
_p~4~_=“I will survive the night."
_p~1~_$\vee$_p~2~_$\rightarrow$_p~3~_$\vee$¬ _p~4~_

d.
_p~1~_="Jan works at A company."
_p~2~_="Bob works at A company."
_p~3~_="Jan certainly works more than Bob."
_p~1~_$\wedge$_p~2~_$\rightarrow$_p~3~_

+ a. I love cats and cats barf.
b. If I love cats, then I do not own a dog.
c. It is either the case that I love cats or if the cat barfs, I would own a dog.
d. If my cat barfs, then I'm sending away my cat and I don't love my cat.
e. If I love dogs, then both I own dogs and I'm sending away my cats.
## Problem 2: Thinking About Proof

For $\top$, since the proposition is always provable, it is provable.
E.g:
_p~1~_="All bachelors are single."
_p~1~_$\top$
It's analytic so it is provable.

For $\bot$, since the proposition is never provable, we just cannot prove it -_-.
E.g:
_p~1~_="There is something faster than light."
_p~1~_$\bot$

For $\neg$, we have to prove the negation of the proposition is provable.
E.g:
_p~1~_="It is not raining now."
$\neg$_p~1~_

For $\wedge$, we have to prove both atomic propositions to prove the conjunction.
E.g:
_p~1~_="Jacob is Chinese"
_p~2~_="Jacob is in Grinnell"
_p~1~_ $\wedge$ _p~2~_

For $\vee$, we only need to prove one side of the atomic propositions to prove the disjunction.
E.g.
_p~1~_="Dhall is great."
_p~2~_="Grill is great."
_p~1~_ $\vee$ _p~2~_
We just need to prove either one of atomic propositon is true to prove that Grinnell College serves good food.

For $\rightarrow$, we have to prove that the proposition after $\rightarrow$ is provable assuming that the proposition before $\rightarrow$ is provable.
E.g.
_p~1~_="PM is in Grinnell."
_p~2~_="PM is in Iowa."
If PM is in Grinnell, then PM is in Iowa.
_p~1~_$\rightarrow$_p~2~_